I have a dumb question about physics. Let's consider the one-dimensional heat equation du/dt = d^2 u / dx^2 with boundary conditions u(0, t) = u(L, t) = 0. An example of a solution is u = e^{-t^2/2} sin (pi x / L).

Now, if I think of this solution as being supported on the interval [0, L], then energy is clearly not conserved. But the Wikipedia article claims that the physical derivation of the heat equation assumes conservation of energy. So what gives? Are the boundary conditions somehow sucking all of the energy away?